# ME306 Dynamics, Spring HW1 Solution Key. AB, where θ is the angle between the vectors A and B, the proof

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1 ME6 Dnms, Spng HW Slutn Ke - Pve, gemetll.e. usng wngs sethes n nltll.e. usng equtns n nequltes, tht V then V. Nte: qunttes n l tpee e vets n n egul tpee e sls. Slutn: Let, Then V V V We wnt t pve tht: V. In the ws: V Hee: V n ut sne sθ s then mplete., whee θ s the ngle etween the vets n, the p V Gphll, ne n see tht the mne lengths mgntues vets n tte s gete thn equl t the length V. - s tvelng n pl pth,, whee pstve -etn pnts t the Est n pstve -etn pnts t the Nth. I the stts t n tvels n the Nth-Est etn, hw wul t hve tvele estw t tvele 5 lmetes Nth? Nte: n hve unts lmetes m. Slutn: sne 5. Hene.8 m. - The stne etween pnts n C s t n etween n C s 4 t, wht s the stne etween n? Slutn: Lw Csnes: C L 4 L 4s6 4 ± L 4 C C sγ, L C 4 L γ6

2 4- The pstn hnges wth tme ng t t st.9t sn t. Wht e the st n sen tme evtves? Nte: the Ctesn eeene me t whh the pstn vet s eee s n netl eeene me,.e. ne tht s nt mvng wth espet t Eth. Nte: these tme evtes epesent, espetvel, the velt n eletn the. v & st t sn t.7t t st Slutn: && sn t sn t t st 5.4t st sn t t st 5.4t st t sn t t sn t t 5- F the uve t t, n t n seth t sle the pstn vet n the tngent vet. Slutn: t, t t.5.5 t 6- I the eletn ptle s gven 4 t 6t, n the eletn s the tme evtve the velt the ptle, n the velt the ptle. Slutn: v t t t t C

3 7- The equtn n ellpse s. Nte tht n ellpse n e thugh t eten nentel n the -etn. Fn the gent vet the sue ese the ellpse. Seth, t sle, the ellpse gent vet t the pnts, n,. Wht es the seth shws? Cn u genele u sevtn the gent n thee-mensnl sue? Slutn: The equtn the sue the ellpse s: F,, The gent the untn F s: F F F F,, { F,, F, F, F, Geneltn: The gent vet sue s lws nml pepenul t ts nugte sue. 8- Evlute:, sn s, e, whee n e nstnts. F Slutn: F C ln F u sn sn { s uu C C F u e u ve, u. e ln C we use ntegtn pts: uv uv vu. Te u n ve e e e e e e e e e e

4 9- Plt sn m t usng n spesheet mthemtl stwe. Clulte sn. Fm the plt, n m the plt nl, n the e une the uve shw ll etls suh lultn, n mpe ths esult t tht n pt ths plem. C ls, m the plt, n the slpe the tngent lne t sn t 4. Cmpe ths slpe t the evtve sn.e. sn when evlute t the sme pnt. D Wht s u nlusn, n, m ng pts n C. Slutn: sn 44 sn [ s ] [ s s] e une the uve s sn s 4 C 4 D The slpe the uve s equl t the evtve sn t the sme pnt. - I 6, n n. Slutn: I t n t, n the pl ntes n θ n tems the pmete t. Plt t sle the pth ese ps n t n the nge [,]. 4t 4t t 4t 4 5t 8t 5 θ tn tn t t 4 4 t 4 5 t 5 4

5 - I s lne untn, pve tht, whee vg s the vege vlue ve the nge [, ]. Illustte epln ths eqult gphll ls. Slutn: [ ] vg vg Q. En P. vg Slpe up vg wn Nte tht the e une the uvestght lne.e. the ntegl, s equl t the she e,.e. vg -. 5

6 6 - Plem 7, pge 659 n u tet. 4- Fn the ngle etween the tw vets n. 4, s s s θ θ 5- Fn unt vet nml t the llwng tw vets: n. Slutn: te the ss put n then nmle the esult. u vet unt

7 6- Plem 4 n tet. 7- Plem 4, pge 658 n tet. 7

8 8 8- Fn eve the ente mss ent the mss enst s unm the llwng tw ets: semul plte us. Cente mss s t, untn 44 m Clulus mmetel m smmet the et ut the -s. [ ] [ ] [ ] [ ] [ ] 4 the llwng tngle: gn, m smmet ut the -s,. [ ] [ ] [ ] [ ] [ ] [ ] [ ] -

9 9- Thee mm-mete sphees nstute eent metls e lte t the vetes n equltel tngle n eep spe. Detemne the esultnt R the gvttnl es whh the lumnum n st-n sphees eet n the ppe sphee. Nte: the stne etween n tw sphees s m. lumnum F F C C Cst In Cppe us.5m ll sphees G ρ ρ 6.67E Gmm F 4.8E N G ρ ρ C 6.67E Gm mc F C.75E 9N C 9 R F FC 4.8 [ sn s ].75 [ sn s ] N 9

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